On the parity of generalized partition functions, III
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 51-78.

Dans cet article, nous complétons les résultats de J.-L. Nicolas [15], en déterminant tous les éléments de l’ensemble 𝒜=𝒜(1+z+z 3 +z 4 +z 5 ) pour lequel la fonction de partition p(𝒜,n) (c-à-d le nombre de partitions de n en parts dans 𝒜) est paire pour tout n6. Nous donnons aussi un équivalent asymptotique à la fonction de décompte de cet ensemble.

Improving on some results of J.-L. Nicolas [15], the elements of the set 𝒜=𝒜(1+z+z 3 +z 4 +z 5 ), for which the partition function p(𝒜,n) (i.e. the number of partitions of n with parts in 𝒜) is even for all n6 are determined. An asymptotic estimate to the counting function of this set is also given.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.704
Classification : 11P81,  11N25,  11N37
Mots clés : Partitions, periodic sequences, order of a polynomial, orbits, 2-adic numbers, counting function, Selberg-Delange formula.
@article{JTNB_2010__22_1_51_0,
     author = {Fethi Ben Sa{\"\i}d and Jean-Louis Nicolas and Ahlem Zekraoui},
     title = {On the parity of generalized partition {functions,~III}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {51--78},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.704},
     zbl = {1236.11088},
     mrnumber = {2675873},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/}
}
TY  - JOUR
AU  - Fethi Ben Saïd
AU  - Jean-Louis Nicolas
AU  - Ahlem Zekraoui
TI  - On the parity of generalized partition functions, III
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2010
DA  - 2010///
SP  - 51
EP  - 78
VL  - 22
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/
UR  - https://zbmath.org/?q=an%3A1236.11088
UR  - https://www.ams.org/mathscinet-getitem?mr=2675873
UR  - https://doi.org/10.5802/jtnb.704
DO  - 10.5802/jtnb.704
LA  - en
ID  - JTNB_2010__22_1_51_0
ER  - 
Fethi Ben Saïd; Jean-Louis Nicolas; Ahlem Zekraoui. On the parity of generalized partition functions, III. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 51-78. doi : 10.5802/jtnb.704. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/

[1] N. Baccar, Sets with even partition functions and 2-adic integers. Periodica Math. Hung. 55 (2) (2007), 177–193. | MR 2375042 | Zbl 1164.11066

[2] N. Baccar and F. Ben Saïd, On sets such that the partition function is even from a certain point on. International Journal of Number Theory 5 n o 3 (2009), 407–428. | MR 2529082 | Zbl 05589358

[3] N. Baccar, F. Ben Saïd and A. Zekraoui, On the divisor function of sets with even partition functions. Acta Math. Hungarica 112 (1-2) (2006), 25–37. | MR 2251128 | Zbl 1121.11071

[4] F. Ben Saïd, On a conjecture of Nicolas-Sárközy about partitions. Journal of Number Theory 95 (2002), 209–226. | MR 1924098 | Zbl 1041.11066

[5] F. Ben Saïd, On some sets with even valued partition function. The Ramanujan Journal 9 (2005), 63–75. | MR 2166378 | Zbl 1145.11073

[6] F. Ben Saïd and J.-L. Nicolas, Sets of parts such that the partition function is even. Acta Arithmetica 106 (2003), 183–196. | EuDML 278790 | MR 1958984 | Zbl 1052.11069

[7] F. Ben Saïd and J.-L. Nicolas, Sur une application de la formule de Selberg-Delange. Colloquium Mathematicum 98 n o 2 (2003), 223–247. | EuDML 284796 | MR 2033110 | Zbl 1051.11049

[8] F. Ben Saïd and J.-L. Nicolas, Even partition functions. Séminaire Lotharingien de Combinatoire 46 (2002), B 46i (http//www.mat.univie.ac.at/ slc/). | EuDML 123676 | MR 1921679 | Zbl 1042.11008

[9] F. Ben Saïd, H. Lahouar and J.-L. Nicolas, On the counting function of the sets of parts such that the partition function takes even values for n large enough. Discrete Mathematics 306 (2006), 1089–1096. | MR 2245637 | Zbl 1109.05019

[10] P. M. Cohn, Algebra, Volume 1, Second Edition. John Wiley and Sons Ltd, 1988). | MR 663370

[11] H. Halberstam and H.-E. Richert, Sieve methods. Academic Press, New York, 1974. | MR 424730 | Zbl 0298.10026

[12] H. Lahouar, Fonctions de partitions à parité périodique. European Journal of Combinatorics 24 (2003), 1089–1096. | MR 2024560 | Zbl 1049.11110

[13] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, revised edition, 1994. | MR 1294139 | Zbl 0820.11072

[14] J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy, On the parity of additive representation functions. J. Number Theory 73 (1998), 292–317. | MR 1657968 | Zbl 0921.11050

[15] J.-L. Nicolas, On the parity of generalized partition functions II. Periodica Mathematica Hungarica 43 (2001), 177–189. | MR 1830575 | Zbl 0980.11049

Cité par Sources :